# Criticality Across the Energy Spectrum from Random, Artificial Gravitational Lensing in Two-Dimensional Dirac Superconductors

## TL;DR

This paper investigates how random gravitational-like disorder affects two-dimensional Dirac materials, revealing a universal quantum-critical state across energy levels, with implications for high-temperature superconductors and topological states.

## Contribution

It introduces a novel model of quenched gravitational disorder in 2D Dirac systems, showing universal quantum-critical behavior and explaining experimental phenomena in cuprate superconductors.

## Key findings

- Disorder induces a quantum-critical 'stack' of states with universal fluctuations.
- Model with pseudospin rotations and nematic fluctuations best matches cuprate phenomenology.
- Energy width of critical states increases with disorder strength.

## Abstract

We numerically study weak, random, spatial velocity modulation ["quenched gravitational disorder" (QGD)] in two-dimensional massless Dirac materials. QGD couples to the spatial components of the stress tensor; the gauge-invariant disorder strength is encoded in the quenched curvature. Although expected to produce negligible effects, wave interference due to QGD transforms all but the lowest-energy states into a quantum-critical "stack" with universal, energy-independent spatial fluctuations. We study five variants of velocity disorder, incorporating three different local deformations of the Dirac cone: flattening or steepening of the cone, pseudospin rotations, and nematic deformation (squishing) of the cone. QGD should arise for nodal excitations in the $d$-wave cuprate superconductors (SCs), due to gap inhomogeneity. Our results may explain the division between low-energy "coherent" (plane-wave-like) and finite-energy "incoherent" (spatially inhomogeneous) excitations in the SC and pseudogap regimes. The model variant that best matches the cuprate phenomenology possesses quenched random pseudospin rotations and nematic fluctuations. This model variant and another with pure nematic randomness exhibit a robust energy swath of stacked critical states, the width of which increases with increasing disorder strength. By contrast, quenched fluctuations that isotropically flatten or steepen the Dirac cone tend to produce strong disorder effects, with more rarified wave functions at low- and high-energies. Our models also describe the surface states of class DIII topological SCs.